Column $\ell_{2,0}$-norm regularized factorization model of low-rank matrix recovery and its computation

This paper is concerned with the column $\ell_{2,0}$-regularized factorization model of low-rank matrix recovery problems and its computation. The column $\ell_{2,0}$-norm of factor matrices is introduced to promote column sparsity of factors and lower rank solutions. For this nonconvex nonsmooth and non-Lipschitz problem, we develop an alternating majorization-minimization (AMM) method with extrapolation, and a hybrid AMM in which a majorized alternating proximal method is first proposed to seek an initial factor pair with less nonzero columns and then the AMM with extrapolation is applied to the minimization of smooth nonconvex loss. We provide the global convergence analysis for the proposed AMM methods and apply them to the matrix completion problem with non-uniform sampling schemes. Numerical experiments are conducted with synthetic and real data examples, and comparison results with the nuclear-norm regularized factorization model and the max-norm regularized convex model demonstrate that the column $\ell_{2,0}$-regularized factorization model has an advantage in offering solutions of lower error and rank within less time.